On Kähler metrisability of two-dimensional complex projective structures

DOI 10.1007/s00605-014-0636-0

On Kähler metrisability of two-dimensional complex

projective structures

Thomas Mettler

Received: 16 April 2013 / Accepted: 6 May 2014 / Published online: 27 May 2014 © Springer-Verlag Wien 2014

Abstract We derive necessary conditions for a complex projective structure on a complex surface to arise via the Levi-Civita connection of a (pseudo-)Kähler metric. Furthermore we show that the (pseudo-)Kähler metrics defined on some domain in the projective plane which are compatible with the standard complex projective structure are in one-to-one correspondence with the hermitian forms onC3whose rank is at least two. This is achieved by prolonging the relevant finite-type first order linear differential system to closed form. Along the way we derive the complex projective Weyl and Liouville curvature using the language of Cartan geometries.

Keywords Complex projective geometry· Cartan geometry · Metrisability Mathematics Subject Classification (2010) 53A20· 53B10

1 Introduction

Recall that an equivalence class of affine torsion-free connections on the tangent bundle of a smooth manifold N is called a (real) projective structure [11,38,39]. Two connections∇ and ∇are projectively equivalent if they share the same unparametrised

Communicated by A. Cap.

The author was supported by Forschungsinstitut für Mathematik (fim) at ETH Zürich and Schweizerischer Nationalfondssnf via the postdoctoral fellowship PA00P2_142053. The author is grateful to Friedrich-Schiller-Universität Jena for financial support for several trips to Jena where a part of the writing for this article took place.

T. Mettler (

B

Department of Mathematics, ETH Zürich, Zurich, Switzerland e-mail: thomas.mettler@math.ethz.ch

geodesics. This condition is equivalent to∇ and ∇inducing the same parallel transport on the projectivised tangent bundlePT N.

It is a natural task to (locally) characterise the projective structures arising via the Levi-Civita connection of a (pseudo-)Riemannian metric. Liouville [25] made the crucial observation that the Riemannian metrics on a surface whose Levi-Civita connection belongs to a given projective class precisely correspond to nondegen-erate solutions of a certain projectively invariant finite-type linear system of partial differential equations. In [3] Bryant, Eastwood and Dunajski used Liouville’s observa-tion to solve the two-dimensional version of the Riemannian metrisability problem. In another direction it was shown in [29] that on a surface locally every affine torsion-free connection is projectively equivalent to a conformal connection (see also [28]). Local existence of a connection with skew-symmetric Ricci tensor in a given projective class was investigated in [36] (see also [23] for a connection to Veronese webs). Liouville’s result generalises to higher dimensions [30] and the corresponding finite-type differ-ential system was prolonged to closed form in [14,30]. Several necessary conditions for Riemann metrisability of a projective structure in dimensions larger than two were given in [33]. See also [8,16] for the role of Einstein metrics in projective geometry.

Now let M be a complex manifold of complex dimension d > 1 with integrable almost complex structure map J . Two affine torsion-free connections∇ and ∇on T M which preserve J are called complex projectively equivalent if they share the same generalised geodesics (for the notion of a curved complex projective structure on Riemann surfaces see [5]). A generalised geodesic is a smoothly immersed curveγ ⊂

M with the property that the 2-plane spanned by ˙γ and J ˙γ is parallel along γ . Complex

projective geometry was introduced by Otsuki and Tashiro [35,37]. Background on the history of complex projective geometry and its recently discovered connection to Hamiltonian 2-forms (see [1] and references therein) may be found in [26].

In the complex setting it is natural to study the Kähler metrisability problem, i.e. try to (locally) characterise the complex projective structures which arise via the Levi-Civita connection of a (pseudo-)Kähler metric. Similar to the real case, the Kähler metrics whose Levi-Civita connection belongs to a given complex projective class precisely correspond to nondegenerate solutions of a certain complex projectively invariant finite-type linear system of partial differential equations [12,26,31].

In this note we prolong the relevant differential system to closed form in the surface case. In doing so we obtain necessary conditions for Kähler metrisability of a complex projective structure[∇] on a complex surface and show in particular that the generic complex projective structure is not Kähler metrisable. Furthermore we show that the space of Kähler metrics compatible with a given complex projective structure is alge-braically constrained by the complex projective Weyl curvature of[∇]. We also show that the (pseudo-)Kähler metrics defined on some domain inCP2which are compatible with the standard complex projective structure are in one-to-one correspondence with the hermitian forms onC3whose rank is at least two. A result whose real counterpart is a well-known classical fact. This note concerns itself with the complex 2-dimensional case, but there are obvious higher dimensional generalisations that can be treated with the same techniques.

The reader should be aware that the results presented here can also be obtained by using the elegant and powerful theory of Bernstein–Gelfand–Gelfand (BGG)

sequences developed by ˇCap, Slovák and Souˇcek [10] (see also the article of Calder-bank and Diemer [6]). In particular, the prolongation computed here is an example of a prolongation connection of a first BGG equation in parabolic geometry and may be derived using the techniques developed in [18].

This note aims at providing an intermediate analysis between the abstract BGG machinery and pure local coordinate computations. This is achieved by carrying out the computations on the parabolic Cartan geometry of a complex projective surface. 2 Complex projective surfaces

2.1 Definitions

Let M be a complex 2-manifold with integrable almost complex structure map J and∇ an affine torsion-free connection on T M. We call ∇ complex-linear if ∇ J = 0. A generalised geodesic for ∇ is a smoothly immersed curve γ ⊂ M with the property that the 2-plane spanned by ˙γ and J ˙γ is parallel along γ , i.e. γ satisfies the reparametrisation invariant condition

∇˙γ˙γ ∧ ˙γ ∧ J ˙γ = 0. (1)

We call two complex linear torsion-free connections∇ and ∇on M complex projec-tively equivalent, if they have the same generalised geodesics. An equivalence class of complex projectively equivalent connections is called a complex projective structure and will be denoted by[∇]. A complex 2-manifold equipped with a complex projective structure will be called a complex projective surface.

Remark 1 What we here call a complex projective structure was originally called a

holomorphic projective structure by Tashiro [37] and others. Once it was realised that in general complex projective structures are not holomorphic in any reasonable way, the name h-projective structure was used—and is still so in very recent times—see for instance [15,21,26]. Furthermore, what we here call generalised geodesics are called h-planar curves in the literature using the name h-projective. One might argue that the notion of a complex projective structure can be confused with well-established notions in algebraic geometry. For this reason complex projective is sometimes also abbreviated to c-projective (see [2]).

Extending∇ to the complexified tangent bundle TCM → M, it follows from the

complex linearity of∇ that for every local holomorphic coordinate system z = (zi) :

U → C2on M there exist unique complex-valued functionsΓ_{j k}i on U , so that ∇∂z j∂zk = Γ

i j k∂zi.

We call the functionsΓ_{j k}i the complex Christoffel symbols of∇. Tashiro showed [37] that two torsion-free complex linear connections∇ and ∇on M are complex projec-tively equivalent if and only if there exists a(1,0)-form β ∈ Ω1,0(M, R) so that

∇ZW − ∇ZW = β(Z)W + β(W)Z (2)

for all(1,0) vector fields Z, W ∈ Γ (T1,0M).

WritingΓ_{j k}i and ˆΓ_{j k}i for the complex Christoffel symbols of∇ and ∇with respect to some holomorphic coordinates z= (zi) and β = βidzi, Eq. (2) translates to

ˆ Γi

j k = Γj ki + δijβk+ δikβj. (3)

Note that formally Eq. (3) is identical to the equation relating two real projectively equivalent connections on a real manifold. In particular, similarly to the real case (see [11,38]), the functions

Πi j k = Γj ki − 1 3  Γl l jδki + Γlklδij  (4) are complex projectively invariant in the sense that they only depend on the coor-dinates z. Moreover, locally[∇] can be recovered from the functions Πi_{j k} and two torsion-free complex linear connections are complex projectively equivalent if and only if they give rise to the same functions Πi_{j k} in every holomorphic coordinate system.

A complex projective structure[∇] is called holomorphic if the functions Πi_{j k} are holomorphic in every holomorphic coordinate system. Gunning [17] obtained rela-tions on characteristic classes of complex manifolds carrying holomorphic projective structures. The condition on a manifold to carry a holomorphic projective structure is particularly restrictive in the case of compact complex surfaces. See also the beau-tiful twistorial interpretation of holomorphic projective surfaces by Hitchin [19] and Remark4.

2.2 Cartan geometry

A complex projective structure admits a description in terms of a normal Cartan geom-etry modelled on complex projective spaceCPn, following the work of Ochiai [34]: see [20] and [40]. The reader unfamiliar with Cartan geometries may consult [9] for a modern introduction. We will restrict to the construction in the complex two-dimensional case.

Let PSL(3, C) act on CP2_{from the left in the obvious way and let P denote the} stabiliser subgroup of the element[1, 0, 0]t ∈ CP2. We have:

Theorem 1 Suppose(M, J, [∇]) is a complex projective surface. Then there exists

(up to isomorphism) a unique real Cartan geometry (π : B → M, θ) of type

(PSL(3, C), P) such that for every local holomorphic coordinate system z = (zi_{) :}

U → C2, there exists a unique sectionσz : U → B satisfying

(σz)∗θ = ⎛ ⎜ ⎝ 0 φ₁0 φ₂0 φ1 0 φ11 φ21 φ2 0 φ12 φ22 ⎞ ⎟ ⎠ (5)

where φi 0= dzi, and φij = Πij kdzk, and φi0= Πi kdzk, with Πi j = ΠilkΠ l j k− ∂Πk i j ∂zk

andΠi_{j k} denote the complex projective invariants with respect to zi defined in (4).

Remark 2 Supposeϕ : (M, J, [∇]) → (M, J, [∇]) is a biholomorphism between

complex projective surfaces identifying the complex projective structures, then there exists a diffeomorphism ˆϕ : B → Bwhich is a P-bundle map coveringϕ and which satisfies ˆϕ∗θ= θ. Conversely, every diffeomorphism Φ : B → Bthat is a P-bundle map and satisfiesΦ∗θ = θ is of the form Φ = ˆϕ for a unique biholomorphism ϕ : M → M_{identifying the complex projective structures.}

Example 1 Let B = PSL(3, C) and let θ denote its Maurer-Cartan form. Setting

M = B/P CP2andπ : PSL(3, C) → CP2the natural quotient projection, one obtains a complex projective structure onCP2 whose generalised geodesics are the smoothly immersed curvesγ ⊂ CP1whereCP1 ⊂ CP2 is any linearly embedded projective line. This is precisely the complex projective structure associated to the Levi-Civita connection of the Fubini-Study metric on CP2. This example satisfies dθ + θ ∧ θ = 0 and is hence called flat.

Let (π : B → M, θ) be the Cartan geometry of a complex projective struc-ture (J, [∇]) on a simply-connected surface M whose Cartan connection satisfies dθ + θ ∧ θ = 0. Then there exists a local diffeomorphism Φ : B → PSL(3, C) pulling back the Maurer-Cartan form of PSL(3, C) to θ and consequently, a local biholomorphismϕ : M → CP2identifying the projective structure on M with the standard flat structure onCP2.

2.3 Bianchi-identities

Theorem1implies that the curvature form = dθ + θ ∧ θ satisfies  = dθ + θ ∧ θ = ⎛ ⎜ ⎝ 0 0₁ 0₂ 0 1₁ 1₂ 0 2₁ 2₂ ⎞ ⎟ ⎠ (6) with 0 i = Liθ01∧ θ 2 0+ Kil¯jθ0l ∧ θ0j,  i k= W i kl¯jθ l 0∧ θ0j

for unique complex-valued functions Li, Kil¯j, and Wkli ¯jon B satisfying Wlil ¯j = 0.

obtain

(σz)∗Wkli _¯j= −

∂Πi kl

∂ ¯zj . (7)

Differentiation of the structure Eq. (6) gives

0= d2θ₀i = W_lki _¯jθ₀l ∧ θ₀k∧ θ₀j, and 0 = d2θ₀0= Ki k¯jθ0i ∧ θ

k

0∧ θ0j which yields the algebraic Bianchi-identities

W_lki _¯j= W_kli _¯j, and Ki k¯j= Kki¯j.

2.3.1 Complex projective Weyl curvature

The identities d2θ_ki = 0 yield κi kl_¯j∧ θ0l∧ θ0j = 0 with κi kl¯j= dWkli ¯j+ Wkli ¯j  θ0 0 + θ00  + Kkl¯jθ0i− Wlsi ¯jθks− Wksi ¯jθls+ Wkls ¯jθsi− Wkli¯sθls

which implies that there exist complex-valued functions W_kli _¯j¯sand W_kli _¯js on B satis-fying W_kli _¯j¯s = W_lki _¯j¯s = W_kli _{¯s ¯j}, W_klk_¯j¯s = W_klk_¯js= 0, W_kli _¯js = W_lki _¯js such that dWkli _¯j=  Wkli _¯js+ δkiKsl¯j+ δilKsk¯j− 3δsiKkl¯j  θs 0+ Wkli _¯j¯sθ0s+ ϕ i kl_¯j (8) where ϕi kl_¯j= −Wkli _¯j  θ0 0+ θ00  + Wi ls_¯jθks+ Wksi _¯jθls − Wkls _¯jθsi+ Wkli_¯sθsj. (9)

Let End0(T M, J) denote the bundle whose fibre at p ∈ M consists of the J-linear endomorphisms of TpM which are complex-traceless. It follows with the structure

Eqs. (6,8,9) and straightforward computations, that there exists a unique(1,1)-form

W on M with values in End0(T M, J) for which

W _∂ ∂zl, ∂ ∂zj  _∂ ∂zk = (σz)∗W i kl¯j ∂ ∂zi = − ∂Πi kl ∂ ¯zj ∂ ∂zi

in every local holomorphic coordinate system z = (zi) on M. Here, as usual, we extend tensor fields on M complex multilinearly to the complexified tangent bundle of M. The bundle-valued 2-form W is called the complex projective Weyl curvature of[∇]. We obtain:

Proposition 1 A complex projective structure[∇] on a complex surface (M, J) is

holomorphic if and only if the complex projective Weyl tensor of[∇] vanishes.

2.3.2 Complex projective Liouville curvature

From d2θ_i0∧ θ₀1∧ θ₀2= 0 one sees after a short computation that

dLi = −4Liθ00+ Ljθij + Li jθ0j+ Li¯jθ0j (10) for unique complex-valued functions Li¯j, Li jon B. Using this last equation it is easy

to check that theπ-semibasic quantity (L1θ1 0+ L2θ02) ⊗  θ1 0⊗ θ02  (11) is invariant under the P right action and thus theπ-pullback of a tensor field λ on

M which is called the complex projective Liouville curvature (see the note of

Liou-ville [24] for the construction ofλ in the real case).

Remark 3 In the case of real projective structures on surfaces, the projective Weyl

curvature vanishes identically. Furthermore, note that contrary to the complex projec-tive Liouville curvature, the complex projecprojec-tive Weyl tensor exists as well in higher dimensions, but also contains(2,0) parts (see [37] for details).

The differential Bianchi-identity (8) implies that if the functions W_kli _¯jvanish iden-tically, then the functions Ki k¯jmust vanish identically as well. We have thus shown:

Proposition 2 A complex projective structure[∇] on a complex surface (M, J) is flat

if and only the complex projective Liouville and Weyl curvature vanish.

Remark 4 In [22] Kobayashi and Ochiai classified compact complex surfaces carrying

flat complex projective structures. More recently Dumitrescu [13] showed among other things that a holomorphic projective structure on a compact complex surface must be flat (see also the results by McKay about holomorphic Cartan geometries [27]).

2.3.3 Further identities

We also obtain

with1 κi k¯j= − dKi k¯j+ 1 2εskLi¯jθ s 0− Ki k¯j  2θ0 0 + θ00  + Ksk¯jθis+ Ksi¯jθks− − Ws i k¯jθ 0 s + Ki k_¯sθs_j.

It follows that there are complex-valued functions Ki k¯jland Kkl¯ı ¯jon B satisfying

Ki k¯jl= Kki¯jl, and Kkl¯ı ¯j= Klk¯ı ¯j= Kkl¯j¯ı such that dKi k¯j= Ki k¯js+ 1 4  εskLi¯j+ εsiLk¯j  θs 0+ Ki k¯j¯sθ0s + ϕi k¯j (12) where ϕi k¯j= −Ki k¯j  2θ0 0+ θ00  + Ksk¯jθis+ Ksi¯jθks− Wi ks ¯jθs0+ Ki k¯sθs_j.

2.4 Complex and generalised geodesics

It is worth explaining how the generalised geodesics of [∇] appear in the Cartan geometry(π : B → M, θ). To this end let G ⊂ P ⊂ PSL(3, C) denote the quotient group of the group of upper triangular matrices of unit determinant modulo its center. The quotient B/G is the total space of a fibre bundle over M whose fibre P/G is diffeomorphic to CP1. In fact, B/G may be identified with the total space of the the complex projectivised tangent bundleτ : P(T1,0M) → M of (M, J). Writing

θ = (θi

j)i, j=0..2, Theorem1 implies that the real codimension 4-subbundle of T B

defined byθ₀2 = θ₁2 = 0 descends to a real rank 2 subbundle E ⊂ T P(T1,0M). The

integral manifolds of E can most conveniently be identified in local coordinates. Let

z = (z1, z2) : U → C2be a local holomorphic coordinate system on M and write φ for the pullback of θ with the unique section σz associated to z in Theorem1. We

obtain a local trivialisation of Cartan’s bundle ϕ : U × P → π−1(U) so that for(z, p) ∈ U × P we have

(ϕ∗θ)(z,p)= (ωP)p+ Ad(p−1) ◦ φz (13)

whereωPdenotes the Maurer-Cartan form of P and Ad the adjoint representation of

PSL(3, C). Consider the Lie group ˜P ⊂ SL(3, C) whose elements are of the form

1 _{We writeε}

det a−1b

0 a

(14) for a ∈ GL(2, C) and bt ∈ C2. By construction, the elements of P are equivalence classes of elements in ˜P where two elements are equivalent if they differ by scalar

multiplication with a complex cube root of 1. The canonical projection ˜P → P will

be denoted byν. Note that a piece N of an integral manifold of E that is contained in τ−1_{(U) is covered by a map}

(z1_{, z}2_{, p) : N → U × ˜P} where p: N → ˜P may be taken to be of the form

p= ⎛ ⎝ 1 (a1)2+(a2)2 0 0 0 a1−a2 0 a2 a1 ⎞ ⎠

for smooth complex-valued functions ai : N → C satisfying (a1)2+ (a2)2= 0.

We first consider the case where N is one-dimensional. We fix a local coordinate t on N . It follows with (13) and straightforward calculations that

 ϕ ◦ (z1_{, z}2_{, ν ◦ p)}∗_θ2 0 = a1˙z2− a2˙z1  (a1)2_{+ (a2)}2 2dt where˙zi denote the derivative of zi with respect to t. Hence we may take

a1= ˙z1 and a2= ˙z2.

Writingβ =ϕ ◦ (z1, z2, ν ◦ p) ∗θ₁2and using (13) again, we compute β =˙z1_¨z2_{− ˙z}2_¨z1₊_˙z1_˙z2_(Π2 21− Π11) + (˙z1 1)2Π112 − (˙z2)2Π121  ˙z1₊ +˙z1_˙z2_(Π2 22− Π12) + (˙z1 1)2Π122 − (˙z2)2Π221  ˙z2 dt (˙z1₎2_{+ (˙z}2₎2. Note that sinceΠ_{i k}i = 0 for k = 1, 2, it follows that β ≡ 0 is equivalent to (z1, z2) satisfying the followingODE system

˙zi_¨zj_{+ Π}j kl˙z

k_˙zl_{= ˙z}j_¨zi_{+ Π}i kl˙z

k_˙zl_{, i, j = 1, 2.}

This last system is easily seen to be equivalent to the system (1). Consequently, the one-dimensional integral manifolds of E are the generalised geodesics of[∇].

Note that in the case of two-dimensional integral manifolds the above computations carry over where t is now a complex parameter, i.e. the two-dimensional integral manifolds are immersed complex curves Y ⊂ M for which ∇_˙Y˙Y is proportional to ˙Y

for some (and hence any)∇ ∈ [∇]. This last condition is equivalent to Y being a totally geodesic immersed complex curve with respect to([∇], J) (c.f. [32, Lemma 4.1]) . A totally geodesic immersed complex curve Y ⊂ M which is maximally extended is called a complex geodesic. Since the complex geodesics are the (maximally extended) two-dimensional integral manifolds of E, they exist only provided that E is integrable. We will next determine the integrability conditions for E. Recall that E ⊂ T P(T1,0M)

is defined by the equationsθ₁2= θ₀2= 0 on B. It follows with the structure equations (6) that dθ2 0 = 0 mod θ02, θ12 and dθ2 1 = W 2 11¯jθ 1 0 ∧ θ0j mod θ 2 0, θ 2 1.

Consequently, E is integrable if and only if W_11¯12 = W_11¯22 = 0. As a consequence of (8) and W2 11¯1= 0 we obtain 0= ϕ_11¯12 = −W_11¯12  θ0 0+ θ00  + W2 1s ¯1θ s 1+ W_{1s ¯1}2 θ1s− W_11¯1s θs2+ W112¯sθsj, which is equivalent to 2W2 12¯1= W 1

11¯1. Using the symmetries of the complex projective Weyl tensor we compute

W_11¯11 = −W_21¯12 = 2W_12¯12 = 2W_21¯12 ,

thus showing that W_11¯11 = W_12¯12 = 0. From this we obtain

0= ϕ1_11¯1= 2W_12¯11 θ₁2− W_11¯12 θ₂1+ W_11¯21 θ₁2. thus implying W1 12¯1= W 2 11¯1= W 1

11¯2= 0. Continuing in this vein allows to conclude that all components of the complex projective Weyl tensor must vanish. We may summarise:

Proposition 3 Let(M, J, [∇]) be a complex projective surface. Then the following

statements are equivalent:

(i) [∇] is holomorphic;

(ii) The complex projective Weyl tensor of[∇] vanishes; (iii) The rank 2 bundle E → P(T1,0M) is Frobenius integrable;

(iv) Every complex line L⊂ T1,0M is tangent to a unique complex geodesic.

Remark 5 The standard flat complex projective structure onCP2is holomorphic and

Remark 6 Note that the integrability conditions for E are a special case of a more

general result obtained by ˇCap in [7]. There it is shown that E is part of an elliptic CR structure of CR dimension and codimension 2, which the complex projective structure induces on P(T1,0_{M). Furthermore, it is also shown that the integrability of E is} equivalent to the holomorphicity of the complex projective surface.

3 Kähler metrisability

In this section we will derive necessary conditions for a complex projective struc-ture [∇] on a complex surface (M, J) to arise via the Levi-Civita connection of a (pseudo-)Kähler metric. There exists a complex projectively invariant linear first order differential operator acting on J -hermitian(2,0) tensor fields on M with weight 1/3, i.e sections of the bundle S2_J(T M)⊗4(T∗M) 1/3. This differential operator has the property that nondegenerate sections in its kernel are in one-to-one correspondence with (pseudo-)Kähler metrics on M whose Levi-Civita connection is compatible with [∇] (see [12,26,31]).

3.1 The differential analysis

We will show that in the surface case, the (pseudo-)Kähler metrics on(M, J, [∇]) whose Levi-Civita connection is compatible with[∇] can equivalently be characterised in terms of a differential system on Cartan’s bundle(π : B → M, θ).

Proposition 4 Suppose the (pseudo-)Kähler metric g is compatible with[∇]. Then,

writingπ∗g = gi¯jθ₀i ◦ θ₀jand setting hi¯j= gi¯j

 g1¯1g2 ¯2− |g1¯2|2 _−2/3 , we have dhi¯j= hi¯j  θ0 0+ θ00  + hi¯sθs_j+ hs¯jθis+ hiεs jθ₀s+ hjεsiθ0s (15)

for some complex-valued functions hi on B. Conversely, suppose there exist

complex-valued functions hi¯j = hj¯ı and hi on B solving (15) and satisfying

h_1¯1h_{2 ¯2}− |h_1¯2|2 = 0, then the symmetric 2-form

hi¯j  h_1¯1h_{2 ¯2}− |h_1¯2|2 ₋₂ θi 0◦ θ0j

is theπ-pullback of a [∇]-compatible (pseudo-)Kähler metric on M.

Proof Let g be a (pseudo-)Kähler metric on(M, J) and write g = gi¯jdzi ◦ dzj for

local holomorphic coordinates z = (z1, z2) : U → C2 on M. Denoting by∇ the Levi-Civita connection of g, the identity∇g = 0 is equivalent to

∂gk¯j ∂zi = gs¯jΓ s i k and ∂gk¯j ∂zı = gk¯sΓ s i j,

where Γ_{j k}i denote the complex Christoffel symbols of∇. Abbreviate G = det gi_¯j, then we obtain ∂G ∂zi = G Γ s si.

Hence, the partial derivative of hk¯j= gk¯jG−2/3with respect to zi becomes

∂hk_¯j ∂zi = gl¯jΓ l i kG−2/3− 2 3gk¯jΓ s siG−2/3= hl¯j Γl i k− 2 3Γ s siδkl  = hl¯j Γl i k− 1 3Γ s siδlk− 1 3Γ s skδli  −1 3hl¯j  Γs siδlk− Γsksδli  .

Note that the last two summands in the last equation are antisymmetric in i, k, so that we may write −1 3hl¯j  Γs siδ l k− Γ s skδ l i  = hjεi k

for unique complex-valued functions hion U . We thus get

∂hk¯j

∂zi = hs¯jΠ s

i k+ hjεi k. (16)

In entirely analogous fashion we obtain ∂hk¯j

∂zı = hk¯sΠ s

i j + hkεi j. (17)

Recall from Theorem 1that the coordinate system z : U → C2 induces a unique sectionσz : U → B of Cartan’s bundle such that

(σz)∗θ00= 0, (σz)∗θ0i = dzi, (σz)∗θi_j = Πi_{j k}dzk. (18)

Consequently, using (16,17,18) we see that (15) is necessary.

Conversely, suppose there exist complex-valued functions hi¯j= hj¯ıand hi on B

solving (15) for which

 h_1¯1h_{2 ¯2}− |h1¯2|2  = 0. Setting gi¯j= hi¯j  h1¯1h2 ¯2− |h1¯2|2 ₋₂ we get dgi¯j= −gi¯j  θ0 0+ ¯θ00  + gi¯sθs_j + gs¯jθis+ gi¯j¯sθ₀s+ gi¯jsθ0s (19)

with gi¯j¯s = (h i_¯jhl_¯s+ hi_¯shl_¯j)εlkhk (h1¯1h2 ¯2− |h1¯2|2)3 , and gi¯jk =(h i_¯jhk_¯s+ hk_¯jhi_¯s)εsuhu (h1¯1h2 ¯2− |h1¯2|2)3 . It follows with (19) that there exists a unique J -Hermitian metric g on M such that π∗_{g= g}

i¯jθ0i◦ θ0j. Choose local holomorphic coordinates z= (z1, z2) : U → C2on

M. By abuse of notation we will write gi¯j, gi¯j¯s, gi¯jkfor the pullback of the respective

functions on B by the sectionσz : U → B associated to z. From (19) we obtain

∂gi¯j ∂zs = gu¯jΠ u i s+ gi¯js = gu¯j  Πu i s+ g¯vugi¯vs  = gu¯j  Πu i s+ δiubs + δsubi = gu¯jΓi su where we write bi = hi¯sεsuhu (h1¯1h2 ¯2− |h1¯2|2)11/3 and Γ_{j k}i = Πi_{j k}+ δi_jbk+ δkibj. Likewise we obtain ∂gi¯j ∂zs = gi¯uΓ u j s.

It follows that there exists a complex-linear connection∇ on U which is contained in[∇] and whose complex Christoffel symbols are given by Γ_{j k}i . By construction, the connection ∇ preserves g and hence must be the Levi-Civita connection of g. Furthermore, ∇ being complex-linear implies that g is Kähler. This completes the

proof.  3.1.1 First prolongation Differentiating (15) yields 0= d2hi¯j= εsiηj∧ θ0s+ εs jηi ∧ θ0s− (hs¯jWisv ¯u+ hi¯sWs_{j u¯v})θ0u∧ θ0v (20) with ηk= dhk+ hk  θ0 0− θ 0 0  − hjθ_kj + εi jhk¯jθ_i0.

This implies that we can write

ηi = ai jθ₀j (21)

for unique complex-valued functions ai j on B. Equations (20) and (21) imply

εkiajl− εl jai k = hj¯sW_{i k ¯l}s − hi¯sWs_{jl ¯k} (22)

h= −1 2ε i j_a i j is real-valued. We get ajl = εjlh− 1 2ε i u_h s¯ıWsjl¯u. and thus dhi = hi  θ0 0 − θ00  + hjθij + hi¯sεslθl0+ εi jh− 1 2ε uv_h s¯uWi js¯v  θj 0. Plugging the formula for ai j back into (22) yields the integrability conditions

hs¯jW_{i k ¯l}s − hi¯sWs_{jl ¯k}= 1 2εl jε uv_h s¯uWi ks¯v− 1 2εkiε uv hu¯sWsjl_¯v

This last equation can be simplified so that we obtain:

Proposition 5 A necessary condition for a complex projective surface(M, J, [∇]) to

be Kähler metrisable is that

hj_¯sW_{i k ¯l}s + hl_¯sW_{i k}s _¯j= hk_¯sWs_jl¯ı+ hi_¯sWs_{jl ¯k} (23)

admits a nondegenerate solution hi_¯j= hj_¯ı.

Remark 7 Note that under suitable constant rank assumptions the system (23) defines

a subbundle of the bundle over M whose sections are hermitian forms on(M, J). For a generic complex projective structure[∇] this subbundle does have rank 0.

3.1.2 Second prolongation We start by computing 0= d2hi ∧ θ01∧ θ02= −  hi¯jεj kLk  θ1 0∧ θ01∧ θ 2 0 ∧ θ02 which is equivalent to h_1¯1h_1¯2 h_2¯1h_{2 ¯2}  · L2 −L1  = 0

which cannot have any solution with(h11h22− |h12|2) = 0 unless L1 = L2 = 0. This shows:

Theorem 2 A necessary condition for a complex projective surface to be Kähler

metrisable is that it is Liouville-flat, i.e. its complex projective Liouville curvature vanishes.

Remark 8 Note that the vanishing of the Liouville curvature is equivalent to requesting

that the curvature ofθ is of type (1,1) only, which agrees with general results in [9]. Assuming henceforth L1= L2= 0 we also get

0= d2hi =  εi jη + ϕi j ∧ θj 0 (24) with η = dh + 2hRe(θ0 0) + 2εi jRe(hiθ0j) − 1 2ε kl_h k¯ıεi jKj s ¯lθ0s and ϕi j = dri j+ ri jθ₀0− rsiθsj − rs jθis− hlWi jl_¯sθ0s+ 1 2ε uv_h i¯uKvs ¯j+ hj¯uKvs¯ı θs 0 where ri j = − 1 2ε uv_h s_¯uW_{i j}s_¯v.

It follows with Cartan’s lemma that there are functions ai j k = ai k jsuch that

εi jη + ϕi j = ai j kθ₀k.

Sinceϕi j is symmetric in i, j, this implies

η = 1 2ε

j i_a i j sθ0.s

Since h is real-valued, we must have εj i ai j s= εuvεklhk_¯uKls_¯v. Concluding, we get dh= −2hRe(θ₀0) + 2εklRe(hlθk0) + 1 2ε i j_εkl_Re(h k¯ıKls¯jθ0s). This completes the prolongation procedure.

Remark 9 Note that further integrability conditions can be derived from (24), we

won’t write these out though. Using Proposition4we obtain:

Theorem 3 Let(M, J, [∇]) be a complex projective surface with Cartan geometry (π : B → M, θ). If U ⊂ B is a connected open set on which there exist functions

hi¯j= hj¯ı, hiand h that satisfy the rank 9 linear system

dhi¯j= 2hi¯jRe(θ00) + hi¯sθs_j+ hs¯jθis+ hiεs jθ₀s+ hjεsiθ0s, dhk= 2ihkIm(θ00) + hlθkl + hk¯ıεi jθ0_j + εklh− 1 2ε i j_h s¯ıWkls ¯j  θl 0, dh= −2hRe(θ00) − 2εlkRe(hlθk0) + 1 2ε i j_εkl_Re(h k¯ıKls¯jθ0s), (25)

and(h1¯1h_{2 ¯2}− |h1¯2|2_{) = 0, then the quadratic form}

g= hi¯jθ i 0◦ θ j 0 (h1¯1h2 ¯2− |h1¯2|2)2

is theπ-pullback to U of a (pseudo-)Kähler metric on π(U) ⊂ M that is compatible

with[∇].

From this we get:

Corollary 1 The Kähler metrics defined on some domain U ⊂ CP2which are

com-patible with the standard complex projective structure onCP2are in one-to-one

cor-respondence with the hermitian forms onC3whose rank is at least two.

Proof Suppose the complex projective structure[∇] has vanishing complex projective

Weyl and Liouville curvature. Then the differential system (25) may be written as

d H+ θ H + Hθ∗= 0 (26) with H= H∗= ⎛ ⎝_−h2h _−h22−h2 hh211 h1 h12 −h11 ⎞ ⎠

where∗denotes the conjugate transpose matrix. Recall that in the flat caseθ = g−1dg for some smooth g: B → PSL(3, C), hence the solutions to (26) are

H = g−1C



g−1

_∗

where C= C∗is a constant hermitian matrix of rank at least two. The statement now

follows immediately with Theorem3. 

Remark 10 On can deduce from Corollary1that a Kähler metric g giving rise to flat

complex projective structures must have constant holomorphic sectional curvature. A result first proved in [37] (in all dimensions).

Remark 11 One can also ask for existence of complex projective structures[∇] whose

degree of mobility is greater than one, i.e. they admit several (non-proportional) com-patible Kähler metrics. In [15] (see also [21]) it was shown that the only closed complex projective manifold with degree of mobility greater than two isCPnwith the projective structure arising via the Fubini-Study metric.

Acknowledgments The author is grateful to V. S. Matveev and S. Rosemann for introducing him to the subject of complex projective geometry through many stimulating discussions, some of which took place during very enjoyable visits to Friedrich-Schiller-Universität in Jena. The author also would like to thank R. L. Bryant for sharing with him his notes [4] which contain the proofs of the counterparts for real projective surfaces of Theorem3and Corollary1. Furthermore, the author also would like to thank the referees for several valuable comments.

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